Currently you own a portfolio comprised of the following three securities. How much of the riskiest security should you sell and replace with risk-free securities if you want your portfolio beta to equal 90 percent of the market beta?
Stock Value/Beta
A 16,400/1.06
B 20,500/1.32
C 18,200/0.98
I don't know how to set this problem up. To find the portfolio Beta, you have to muliply by the weights. But if take money from B to put in T-Bills, the weights change. Is this just a trial and error question?
To find out how much of the riskiest security you should sell and replace with risk-free securities in order to achieve a target portfolio beta, you need to set up an equation and solve for the unknown.
Let's denote the amount of money you should sell from the riskiest security as X. Since you want your portfolio beta to be 90 percent of the market beta, the equation for the portfolio beta is:
Portfolio Beta = (Weight of Stock A * Beta of Stock A) + (Weight of Stock B * Beta of Stock B) + (Weight of Stock C * Beta of Stock C)
Given that you are selling X amount of money from Stock B, the weights change. The updated weights are calculated as the original value of each stock minus X divided by the total portfolio value minus X:
Weight of Stock A = (Value of Stock A) / (Total Portfolio Value - X)
Weight of Stock B = (Value of Stock B - X) / (Total Portfolio Value - X)
Weight of Stock C = (Value of Stock C) / (Total Portfolio Value - X)
Now, substitute these updated weights into the portfolio beta equation:
0.90 * Market Beta = [ (Value of Stock A) / (Total Portfolio Value - X) ] * Beta of Stock A + [ (Value of Stock B - X) / (Total Portfolio Value - X) ] * Beta of Stock B + [ (Value of Stock C) / (Total Portfolio Value - X) ] * Beta of Stock C
You can rearrange this equation to solve for X, the amount you need to sell from Stock B:
[ (Value of Stock A) / (Total Portfolio Value - X) ] * Beta of Stock A + [ (Value of Stock B - X) / (Total Portfolio Value - X) ] * Beta of Stock B + [ (Value of Stock C) / (Total Portfolio Value - X) ] * Beta of Stock C = 0.90 * Market Beta
Now you can solve this equation for X using an algebraic solver or by employing trial and error. You want to find the value of X that satisfies the equation and gives you the desired portfolio beta.
Please note that the weights in the equation may need to be adjusted depending on the specifics of the problem, such as whether there are any additional investments or if there are any constraints on the amount you can sell from a particular security.